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Licensed Unlicensed Requires Authentication Published by De Gruyter June 26, 2013

Immersed spheres of finite total curvature into manifolds

Andrea Mondino and Tristan Rivière

Abstract

We prove that a sequence of possibly branched, weak immersions of the 2-sphere 𝕊2 into an arbitrary compact Riemannian manifold (Mm,h) with uniformly bounded area and uniformly bounded L2-norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of 𝕊2 and whose image is made of a connected union of finitely many, possibly branched, weak immersions of 𝕊2 with finite total curvature. We prove moreover that if the sequence belongs to a class γ of π2(Mm), the limiting Lipschitz mapping of 𝕊2 realizes this class as well.

Funding source: ERC

Award Identifier / Grant number: “Geometric Measure Theory in Non-Euclidean Spaces”

This work was written while the first author was visiting the Forschungsinstitut für Mathematik at the ETH Zürich. He would like to thank the Institut for the hospitality and the excellent working conditions.

Received: 2012-3-26
Revised: 2013-6-6
Accepted: 2013-6-17
Published Online: 2013-6-26
Published in Print: 2014-10-1

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